# mlogit: Multinomial logit model In mlogit: Multinomial Logit Models

## Description

Estimation by maximum likelihood of the multinomial logit model, with alternative-specific and/or individual specific variables.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24``` ```mlogit( formula, data, subset, weights, na.action, start = NULL, alt.subset = NULL, reflevel = NULL, nests = NULL, un.nest.el = FALSE, unscaled = FALSE, heterosc = FALSE, rpar = NULL, probit = FALSE, R = 40, correlation = FALSE, halton = NULL, random.nb = NULL, panel = FALSE, estimate = TRUE, seed = 10, ... ) ```

## Arguments

 `formula` a symbolic description of the model to be estimated, `data` the data: an `mlogit.data` object or an ordinary `data.frame`, `subset` an optional vector specifying a subset of observations for `mlogit`, `weights` an optional vector of weights, `na.action` a function which indicates what should happen when the data contains `NA`s, `start` a vector of starting values, `alt.subset` a vector of character strings containing the subset of alternative on which the model should be estimated, `reflevel` the base alternative (the one for which the coefficients of individual-specific variables are normalized to 0), `nests` a named list of characters vectors, each names being a nest, the corresponding vector being the set of alternatives that belong to this nest, `un.nest.el` a boolean, if `TRUE`, the hypothesis of unique elasticity is imposed for nested logit models, `unscaled` a boolean, if `TRUE`, the unscaled version of the nested logit model is estimated, `heterosc` a boolean, if `TRUE`, the heteroscedastic logit model is estimated, `rpar` a named vector whose names are the random parameters and values the distribution : `'n'` for normal, `'l'` for log-normal, `'t'` for truncated normal, `'u' ` for uniform, `probit` if `TRUE`, a multinomial porbit model is estimated, `R` the number of function evaluation for the gaussian quadrature method used if `heterosc = TRUE`, the number of draws of pseudo-random numbers if `rpar` is not `NULL`, `correlation` only relevant if `rpar` is not `NULL`, if true, the correlation between random parameters is taken into account, `halton` only relevant if `rpar` is not `NULL`, if not `NULL`, halton sequence is used instead of pseudo-random numbers. If `halton = NA`, some default values are used for the prime of the sequence (actually, the primes are used in order) and for the number of elements droped. Otherwise, `halton` should be a list with elements `prime` (the primes used) and `drop` (the number of elements droped). `random.nb` only relevant if `rpar` is not `NULL`, a user-supplied matrix of random, `panel` only relevant if `rpar` is not `NULL` and if the data are repeated observations of the same unit ; if `TRUE`, the mixed-logit model is estimated using panel techniques, `estimate` a boolean indicating whether the model should be estimated or not: if not, the `model.frame` is returned, `seed` the seed to use for random numbers (for mixed logit and probit models), `...` further arguments passed to `mlogit.data` or `mlogit.optim`.

## Details

For how to use the formula argument, see `Formula()`.

The `data` argument may be an ordinary `data.frame`. In this case, some supplementary arguments should be provided and are passed to `mlogit.data()`. Note that it is not necessary to indicate the choice argument as it is deduced from the formula.

The model is estimated using the `mlogit.optim()`. function.

The basic multinomial logit model and three important extentions of this model may be estimated.

If `heterosc=TRUE`, the heteroscedastic logit model is estimated. `J - 1` extra coefficients are estimated that represent the scale parameter for `J - 1` alternatives, the scale parameter for the reference alternative being normalized to 1. The probabilities don't have a closed form, they are estimated using a gaussian quadrature method.

If `nests` is not `NULL`, the nested logit model is estimated.

If `rpar` is not `NULL`, the random parameter model is estimated. The probabilities are approximated using simulations with `R` draws and halton sequences are used if `halton` is not `NULL`. Pseudo-random numbers are drawns from a standard normal and the relevant transformations are performed to obtain numbers drawns from a normal, log-normal, censored-normal or uniform distribution. If `correlation = TRUE`, the correlation between the random parameters are taken into account by estimating the components of the cholesky decomposition of the covariance matrix. With G random parameters, without correlation G standard deviations are estimated, with correlation G * (G + 1) /2 coefficients are estimated.

## Value

An object of class `"mlogit"`, a list with elements:

• coefficients: the named vector of coefficients,

• logLik: the value of the log-likelihood,

• hessian: the hessian of the log-likelihood at convergence,

• gradient: the gradient of the log-likelihood at convergence,

• call: the matched call,

• est.stat: some information about the estimation (time used, optimisation method),

• freq: the frequency of choice,

• residuals: the residuals,

• fitted.values: the fitted values,

• formula: the formula (a `Formula` object),

• expanded.formula: the formula (a `formula` object),

• model: the model frame used,

• index: the index of the choice and of the alternatives.

Yves Croissant

## References

\insertRef

MCFA:73mlogit

\insertRef

MCFA:74mlogit

\insertRef

TRAI:09mlogit

`mlogit.data()` to shape the data. `nnet::multinom()` from package `nnet` performs the estimation of the multinomial logit model with individual specific variables. `mlogit.optim()` details about the optimization function.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81``` ```## Cameron and Trivedi's Microeconometrics p.493 There are two ## alternative specific variables : price and catch one individual ## specific variable (income) and four fishing mode : beach, pier, boat, ## charter data("Fishing", package = "mlogit") Fish <- dfidx(Fishing, varying = 2:9, shape = "wide", choice = "mode") ## a pure "conditional" model summary(mlogit(mode ~ price + catch, data = Fish)) ## a pure "multinomial model" summary(mlogit(mode ~ 0 | income, data = Fish)) ## which can also be estimated using multinom (package nnet) summary(nnet::multinom(mode ~ income, data = Fishing)) ## a "mixed" model m <- mlogit(mode ~ price + catch | income, data = Fish) summary(m) ## same model with charter as the reference level m <- mlogit(mode ~ price + catch | income, data = Fish, reflevel = "charter") ## same model with a subset of alternatives : charter, pier, beach m <- mlogit(mode ~ price + catch | income, data = Fish, alt.subset = c("charter", "pier", "beach")) ## model on unbalanced data i.e. for some observations, some ## alternatives are missing # a data.frame in wide format with two missing prices Fishing2 <- Fishing Fishing2[1, "price.pier"] <- Fishing2[3, "price.beach"] <- NA mlogit(mode ~ price + catch | income, Fishing2, shape = "wide", varying = 2:9) # a data.frame in long format with three missing lines data("TravelMode", package = "AER") Tr2 <- TravelMode[-c(2, 7, 9),] mlogit(choice ~ wait + gcost | income + size, Tr2) ## An heteroscedastic logit model data("TravelMode", package = "AER") hl <- mlogit(choice ~ wait + travel + vcost, TravelMode, heterosc = TRUE) ## A nested logit model TravelMode\$avincome <- with(TravelMode, income * (mode == "air")) TravelMode\$time <- with(TravelMode, travel + wait)/60 TravelMode\$timeair <- with(TravelMode, time * I(mode == "air")) TravelMode\$income <- with(TravelMode, income / 10) # Hensher and Greene (2002), table 1 p.8-9 model 5 TravelMode\$incomeother <- with(TravelMode, ifelse(mode %in% c('air', 'car'), income, 0)) nl <- mlogit(choice ~ gcost + wait + incomeother, TravelMode, nests = list(public = c('train', 'bus'), other = c('car','air'))) # same with a comon nest elasticity (model 1) nl2 <- update(nl, un.nest.el = TRUE) ## a probit model ## Not run: pr <- mlogit(choice ~ wait + travel + vcost, TravelMode, probit = TRUE) ## End(Not run) ## a mixed logit model ## Not run: rpl <- mlogit(mode ~ price + catch | income, Fishing, varying = 2:9, rpar = c(price= 'n', catch = 'n'), correlation = TRUE, alton = NA, R = 50) summary(rpl) rpar(rpl) cor.mlogit(rpl) cov.mlogit(rpl) rpar(rpl, "catch") summary(rpar(rpl, "catch")) ## End(Not run) # a ranked ordered model data("Game", package = "mlogit") g <- mlogit(ch ~ own | hours, Game, varying = 1:12, ranked = TRUE, reflevel = "PC", idnames = c("chid", "alt")) ```

### Example output

```Loading required package: dfidx

Attaching package: ‘dfidx’

The following object is masked from ‘package:stats’:

filter

Call:
mlogit(formula = mode ~ price + catch, data = Fish, method = "nr")

Frequencies of alternatives:choice
beach    boat charter    pier
0.11337 0.35364 0.38240 0.15059

nr method
7 iterations, 0h:0m:0s
g'(-H)^-1g = 6.22E-06
successive function values within tolerance limits

Coefficients :
Estimate Std. Error  z-value  Pr(>|z|)
(Intercept):boat     0.8713749  0.1140428   7.6408 2.154e-14 ***
(Intercept):charter  1.4988884  0.1329328  11.2755 < 2.2e-16 ***
(Intercept):pier     0.3070552  0.1145738   2.6800 0.0073627 **
price               -0.0247896  0.0017044 -14.5444 < 2.2e-16 ***
catch                0.3771689  0.1099707   3.4297 0.0006042 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log-Likelihood: -1230.8
Likelihood ratio test : chisq = 533.88 (p.value = < 2.22e-16)

Call:
mlogit(formula = mode ~ 0 | income, data = Fish, method = "nr")

Frequencies of alternatives:choice
beach    boat charter    pier
0.11337 0.35364 0.38240 0.15059

nr method
4 iterations, 0h:0m:0s
g'(-H)^-1g = 8.32E-07
gradient close to zero

Coefficients :
Estimate  Std. Error z-value  Pr(>|z|)
(Intercept):boat     7.3892e-01  1.9673e-01  3.7560 0.0001727 ***
(Intercept):charter  1.3413e+00  1.9452e-01  6.8955 5.367e-12 ***
(Intercept):pier     8.1415e-01  2.2863e-01  3.5610 0.0003695 ***
income:boat          9.1906e-05  4.0664e-05  2.2602 0.0238116 *
income:charter      -3.1640e-05  4.1846e-05 -0.7561 0.4495908
income:pier         -1.4340e-04  5.3288e-05 -2.6911 0.0071223 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log-Likelihood: -1477.2
Likelihood ratio test : chisq = 41.145 (p.value = 6.0931e-09)
# weights:  12 (6 variable)
initial  value 1638.599935
iter  10 value 1477.150646
final  value 1477.150569
converged
Call:
nnet::multinom(formula = mode ~ income, data = Fishing)

Coefficients:
(Intercept)        income
pier      0.8141506 -1.434028e-04
boat      0.7389178  9.190824e-05
charter   1.3412901 -3.163844e-05

Std. Errors:
(Intercept)       income
pier    5.816490e-09 2.668383e-05
boat    3.209473e-09 2.057825e-05
charter 3.921689e-09 2.116425e-05

Residual Deviance: 2954.301
AIC: 2966.301

Call:
mlogit(formula = mode ~ price + catch | income, data = Fish,
method = "nr")

Frequencies of alternatives:choice
beach    boat charter    pier
0.11337 0.35364 0.38240 0.15059

nr method
7 iterations, 0h:0m:0s
g'(-H)^-1g = 1.37E-05
successive function values within tolerance limits

Coefficients :
Estimate  Std. Error  z-value  Pr(>|z|)
(Intercept):boat     5.2728e-01  2.2279e-01   2.3667 0.0179485 *
(Intercept):charter  1.6944e+00  2.2405e-01   7.5624 3.952e-14 ***
(Intercept):pier     7.7796e-01  2.2049e-01   3.5283 0.0004183 ***
price               -2.5117e-02  1.7317e-03 -14.5042 < 2.2e-16 ***
catch                3.5778e-01  1.0977e-01   3.2593 0.0011170 **
income:boat          8.9440e-05  5.0067e-05   1.7864 0.0740345 .
income:charter      -3.3292e-05  5.0341e-05  -0.6613 0.5084031
income:pier         -1.2758e-04  5.0640e-05  -2.5193 0.0117582 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log-Likelihood: -1215.1
Likelihood ratio test : chisq = 565.17 (p.value = < 2.22e-16)

Call:
mlogit(formula = mode ~ price + catch | income, data = Fishing2,     shape = "wide", varying = 2:9, method = "nr")

Coefficients:
(Intercept):boat  (Intercept):charter     (Intercept):pier
5.2790e-01           1.6948e+00           7.7663e-01
price                catch          income:boat
-2.5110e-02           3.5768e-01           8.9122e-05
income:charter          income:pier
-3.3611e-05          -1.2700e-04

Call:
mlogit(formula = choice ~ wait + gcost | income + size, data = Tr2,     method = "nr")

Coefficients:
(Intercept):train    (Intercept):bus    (Intercept):car               wait
-2.3115942         -3.4504941         -7.8913907         -0.1013180
gcost       income:train         income:bus         income:car
-0.0197064         -0.0589804         -0.0277037         -0.0041153
size:train           size:bus           size:car
1.3289497          1.0090796          1.0392585

Call:
mlogit(formula = mode ~ price + catch | income, data = Fishing,
rpar = c(price = "n", catch = "n"), R = 50, correlation = TRUE,
varying = 2:9, alton = NA)

Frequencies of alternatives:choice
beach    boat charter    pier
0.11337 0.35364 0.38240 0.15059

bfgs method
17 iterations, 0h:0m:9s
g'(-H)^-1g = 1.3E-07
gradient close to zero

Coefficients :
Estimate  Std. Error z-value  Pr(>|z|)
(Intercept):boat     3.4903e-01  2.6558e-01  1.3142 0.1887779
(Intercept):charter  2.1865e+00  3.0504e-01  7.1680 7.612e-13 ***
(Intercept):pier     7.4647e-01  2.1246e-01  3.5135 0.0004422 ***
price               -4.4544e-02  4.9868e-03 -8.9325 < 2.2e-16 ***
catch                2.8054e-01  1.5687e-01  1.7884 0.0737120 .
income:boat          1.0513e-04  5.8634e-05  1.7929 0.0729872 .
income:charter      -4.0786e-05  5.9634e-05 -0.6839 0.4940114
income:pier         -1.2163e-04  4.7312e-05 -2.5708 0.0101451 *
chol.price:price     2.3047e-02  4.1211e-03  5.5925 2.238e-08 ***
chol.price:catch     7.5142e-01  3.6695e-01  2.0477 0.0405860 *
chol.catch:catch     1.3290e-02  7.8229e-01  0.0170 0.9864459
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log-Likelihood: -1185.9
Likelihood ratio test : chisq = 623.63 (p.value = < 2.22e-16)

random coefficients
Min.     1st Qu.      Median        Mean     3rd Qu. Max.
price -Inf -0.06008954 -0.04454446 -0.04454446 -0.02899938  Inf
catch -Inf -0.22636264  0.28054193  0.28054193  0.78744649  Inf
\$price
normal distribution with parameters -0.045 (mean) and 0.023 (sd)

\$catch
normal distribution with parameters 0.281 (mean) and 0.752 (sd)

price     catch
price 1.0000000 0.9998436
catch 0.9998436 1.0000000
price      catch
price 0.0005311717 0.01731811
catch 0.0173181068 0.56480911
normal distribution with parameters 0.281 (mean) and 0.752 (sd)
Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
-Inf -0.2263626  0.2805419  0.2805419  0.7874465        Inf
```

mlogit documentation built on Oct. 23, 2020, 5:29 p.m.